Independence in computable algebra

Abstract

We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and difference closed fields with the relevant notions of independence. To cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind. We will then apply our techniques to derive new corollaries on the number of computable presentations of these structures. The condition also implies classical and new results on vector spaces, algebraically closed fields, torsion-free abelian groups and Archimedean ordered abelian groups.1991 Mathematics Subject Classification. 03D45, 03C57, 12Y05.

Computable topological abelian groups

“…It replaces the clever combinatorial strategy of Nurtazin [43] with an application of abelian group theory. The result then becomes a special case of the more general theorem from [24] which is not restricted to groups but works for other classes too (such as differential closed fields, for instance).…”

“…We thus can safely dispose of g k(s) by declaring it equal to a linear combination of the generators which have not yet been declared zero using sufficiently large coefficients. For a modern clarification and verification of this strategy, see the proof of Claim 4.7 in [24]. Note that for the process to eventually stabilise, we need to have at least one non-zero element in U .…”

“…Unfortunately, we cannot use the abstract techniques of [38,24] to extend the theorem of Dobritsa to the t.d.l.c. case; this is because we have to simultaneously approximate the projections between the B i , so the group-theoretic structure on B i cannot be separated from the process of building a maximal linearly independent set.…”

Computable valued fields

“…Many other structures have been found to have the same property, such as algebraically closed fields, torsion-free abelian groups [Nur74,Dob83,Gon82], Archimedean ordered abelian groups [GLS03], differentially closed fields, real closed fields, and difference closed fields [HTMM15]. In [HTMM15], the author together with Melnikov and Montalbán formally characterized this phenomenon (which they named the Mal'cev property) using the notion of a r.i.c.e. pregeometry, and presented a metatheorem unifying all of these examples.…”

“…The metatheorem is stated using the general notion of a pregeometry, but for the purposes of this paper, the pregeometry will always be algebraic independence in fields, and the reader need not know the general definition of a pregeometry. In [HTMM15], two conditions were isolated which imply the Mal'cev property. We require some definitions before we state these conditions and the metatheorem.…”